tors and in Fig. 3-28 have equal
magnitudes of 10.0 m and the angles
are 30° and 105°. Find the
(a)xand (b) ycomponents of their
vector sum , (c) the magnitude of ,
and (d) the angle makes with the
positive direction of the xaxis.
•16 For the displacement vectors
and
, give in
(a) unit-vector notation, and as (b) a
magnitude and (c) an angle (rela-
tive to ). Now give in (d) unit-vector notation, and as (e) a
magnitude and (f) an angle.
•17 Three vectors , , and each have a magnitude of
50 m and lie in an xyplane. Their directions relative to the positive
direction of the xaxis are 30°, 195°, and 315°, respectively. What are
(a) the magnitude and (b) the angle of the vector , and
(c) the magnitude and (d) the angle of? What are the
(e) magnitude and (f) angle of a fourth vector such that
?
•18 In the sum , vector has a magnitude of 12.0 m
and is angled 40.0° counterclockwise from the direction, and vec-
tor has a magnitude of 15.0 m and is angled 20.0° counterclock-
wise from the direction. What are (a) the magnitude and (b) the
angle (relative to ) of?
•19 In a game of lawn chess, where pieces are moved between
the centers of squares that are each 1.00 m on edge, a knight is
moved in the following way: (1) two squares forward, one square
rightward; (2) two squares leftward, one square forward; (3) two
squares forward, one square leftward. What are (a) the magnitude
and (b) the angle (relative to “forward”) of the knight’s overall dis-
placement for the series of three moves?B
:
xxC
: xA
:
A
:
B
:
C
:(a:b
:
)(:cd
:
) 0da :
::b:ca:b
:
:cb :c
:
ILW :ab
:
iˆ :a:ab
:
(5.0 m)iˆ(2.0 m)jˆb
:
:a(3.0 m)iˆ(4.0 m)jˆ :r:r :r 1 2 b
:
a:Module 3-1 Vectors and Their Components
•1 What are (a) the xcomponent and (b) the ycomponent of a
vector in the xyplane if its direction is 250°
counterclockwise from the positive direction
of the xaxis and its magnitude is 7.3 m?
•2 A displacement vector in the xyplane
is 15 m long and directed at angle u30° in
Fig. 3-26. Determine (a) the xcomponent
and (b) the ycomponent of the vector.
•3 The xcomponent of vector is
25.0 m and the ycomponent is 40.0 m. (a) What is the magni-
tude of? (b) What is the angle between the direction of and
the positive direction of x?
•4 Express the following angles in radians: (a) 20.0°, (b) 50.0°,
(c) 100°. Convert the following angles to degrees: (d) 0.330 rad,
(e) 2.10 rad, (f) 7.70 rad.
•5 A ship sets out to sail to a point 120 km due north. An unex-
pected storm blows the ship to a point 100 km due east of its
starting point. (a) How far and (b) in what direction must it now
sail to reach its original destination?
•6 In Fig. 3-27, a heavy piece of
machinery is raised by sliding it a
distance d12.5 m along a plank
oriented at angle u20.0° to the
horizontal. How far is it moved
(a) vertically and (b) horizontally?
•7 Consider two displacements,
one of magnitude 3 m and another
of magnitude 4 m. Show how the
displacement vectors may be combined to get a resultant displace-
ment of magnitude (a) 7 m, (b) 1 m, and (c) 5 m.
Module 3-2 Unit Vectors, Adding Vectors by Components
•8 A person walks in the following pattern: 3.1 km north, then
2.4 km west, and finally 5.2 km south. (a) Sketch the vector dia-
gram that represents this motion. (b) How far and (c) in what di-
rection would a bird fly in a straight line from the same starting
point to the same final point?
•9 Two vectors are given by
and.
In unit-vector notation, find (a) , (b) , and (c) a third
vector such that.
•10 Find the (a) x, (b) y, and (c) zcomponents of the sum of
the displacements and whose components in meters are
cx 7.4,cy 3.8,cz 6.1;dx 4.4,dy 2.0,dz 3.3.
•11 (a) In unit-vector notation, what is the sum if
(4.0 m) (3.0 m) and ( 13.0 m) (7.0 m)? What
are the (b) magnitude and (c) direction of a:b?
: j
:a ˆi jˆ :b ˆi ˆ
:ab
:
SSM d
:
:c:r:ab
:
:c :c 0a:b
:
a:b
:b
:
(1.0 m)iˆ(1.0 m)jˆ(4.0 m)kˆ:a(4.0 m)iˆ(3.0 m)jˆ(1.0 m)kˆA
:
A:
A
:
SSM:r:aSSMPROBLEMS 57θdFigure 3-27Problem 6.•12 A car is driven east for a distance of 50 km, then north for 30
km, and then in a direction 30° east of north for 25 km. Sketch the
vector diagram and determine (a) the magnitude and (b) the angle
of the car’s total displacement from its starting point.
•13 A person desires to reach a point that is 3.40 km from her
present location and in a direction that is 35.0° north of east.
However, she must travel along streets that are oriented either
north – south or east – west. What is the minimum distance she
could travel to reach her destination?
•14 You are to make four straight-line moves over a flat desert
floor, starting at the origin of an xycoordinate system and ending
at the xycoordinates (140 m, 30 m). The xcomponent and y
component of your moves are the following, respectively, in me-
ters: (20 and 60), then (bxand70), then (20 and cy), then ( 60
and70). What are (a) component bxand (b) component cy?
What are (c) the magnitude and (d) the angle (relative to the pos-
itive direction of the xaxis) of the overall displacement?
•15 SSM ILW WWW The two vec-θ
xyrFigure 3-26
Problem 2.θO xy2θ 1abFigure 3-28Problem 15.Tutoring problem available (at instructor’s discretion) in WileyPLUSand WebAssign
SSM Worked-out solution available in Student Solutions Manual- –••• Number of dots indicates level of problem difficulty
Additional information available in The Flying Circus of Physicsand at flyingcircusofphysics.com
WWWWorked-out solution is at
ILW Interactive solution is at http://www.wiley.com/college/hallidayProblems